how to Apply mathematical induction method to prove

Question : Apply mathematical induction method to prove 1.1!+2.2!+3.3!+….+n.(n+1)!=(n+1)!-1

Answer :

Proof by mathematical induction method Let the statement as [P_n:\ 1\cdot1!+2\cdot2!+3\cdot3!+…+n\cdot n!] Base Step : n=1 1.1!=2!-1 =>1=2-1 =>1=1 Thus P(1) is true Induction Step : Let the statement is true for n =k. That is [P\left(k\right):\ 1\cdot1!+2\cdot2!+3\cdot3!+\ldots+k\cdot k!=\left(k+1\right)!-1] Now we to prove the statement for n=k+1 [=1\cdot1!+2\cdot2!+3\cdot3!+\ldots+k\cdot k!+\left(k+1\right)\left(k+1\right)!] [=\left(k+1\right)!-1+\left(k+1\right)\left(k+1\right)!] [=\left(k+1\right)!\left(1+k+1\right)-1] [=\left(k+1\right)!\left(k+2\right)-1] [=\left(k+2\right)!-1] Thus the statement is true for n=k+1. Hence the statement P(n) is true for all n.